• To study the characteristics of speech using linear prediction (LP) analysis.

• To observe the LP spectrum and LP residual for voiced and unvoiced segments.

• To study the effect of order of LP analysis (normalized error), autocorrelation of signal and LP residual for voiced and unvoiced segments.

• To study the glottal pulse characteristics.

Source-system modeling of speech signals using LP analysis

The vocal tract system can be modeled as a time-varying all-pole filter using segmental analysis. The segmental analysis corresponds to the processing of speech as short (10-30 ms) overlapped (5-15 ms) windows. The vocal tract system is assumed to be stationary within the window and is modeled as an all-pole filter of order \( p \) using linear prediction (LP) analysis. The LP analysis works on the principle that a sample value in a correlated, stationary sequence can be predicted as a linear weighted sum of the past few (\( p \)) samples. If \( s(n) \) denotes a sequence of speech samples, then the predicted value at the time instant \( n \) is given by $$ \hat{s}(n) = \sum_{k=1}^{p}{a_k~s(n-k)} \qquad (1) $$ where \( \{a_k\},~k=1,2,...,p \) is the set of linear predictor coefficients (LPC) and \(p\) is the order of the LP filter. The error at time \(n\) and the sum of squared errors \( E \) are given by $$ r(n)~=~s(n)~-~\hat{s}(n) \qquad(2)$$ $$ E=~\sum_{n}{r^2(n)} \qquad(3) $$ The cost function \( E \) is minimized with respect to \( \{a_i\},~i=1,2,...,p \) over the interval \( {-\infty}~{\leq}~n~{\leq}~{\infty} \) (autocorrelation formulation) as, $$ {\partial{E}}/{\partial{a_i}}~=~0~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~1~{\leq}~i~{\leq}~p \qquad(4)$$ This minimization leads to a set of normal equations, $$ \sum_{k=1}^{p}{a_k~R(i-k)} = -R(i)~~~~~~~~~~~~~~1~{\leq}~i~{\leq}~p \qquad(5)$$ where $$ R(i) = \sum_{n=-\infty}^{\infty}s(n)~s(n+i)~~~~~~~~~~-{\infty}~{\leq}~i~{\leq}~{\infty} \qquad(6) $$ is the autocorrelation sequence. The solution of these normal equations gives the values of the predictor coefficients \( \{a_k\},~k=1,2,...,p \). The error signal \( r(n) \) obtained by inverse filtering the speech signal is referred to as the LP residual. The smooth variations (highly correlated) in the speech signal are captured by the LPCs and are attributed to the vocal tract characteristics. The complex poles of the LP filter occur as conjugate pairs, and each pair represents a resonator cavity, with a maximum response at a frequency (called as resonant frequency) where the poles are located on the z-plane. The vocal tract can be considered as a cascade of resonator cavities with different shapes and sizes. The resonant frequencies of these cavities are referred to as formants. The LP residual signal has large error values at regular intervals and can be attributed to the periodic impulses of excitation. Hence the LP residual is a good approximation to the excitation source signal and can be used further to extract the excitation source characteristics. A segment of voiced speech (windowed), frequency response of the inverse filter and the corresponding LP residual are shown in Figure 1.

Figure 1: Inverse filtering the speech signal for estimating the excitation source (LP residual) signal.

Short time spectrum, LP spectrum and Inverse spectrum

The short-time spectrum consists of range of frequencies (magnitude and phase components) that are present in a small segment (10-30 ms) of a signal. An inverse spectrum \(A(j\omega)\) is an all-zero model derived from \(A(z)\). The LP spectrum is an all-pole model given by \(\frac{1}{A(j\omega)}\). The LP spectrum can approximate the envelope of the short-time spectrum depending on the choice of LP order. The gross envelope is captured at low LP orders such as 1 or 3. The short-time spectral envelope is finely matched as the LP order increases. One of the main issues in LP analysis is the choice of appropriate LP order. Figures 2 and 3 show the short-time spectrum, LP spectrum and the inverse LP spectrum for a segment of voiced (/a/) and unvoiced speech (/s/) respectively.

Figure 2: (a) Segment of voiced speech /a/ and its (b) short time spectrum, (c) LP spectrum and (d) inverse spectrum (LP order: 10).

Figure 3: (a) Segment of unvoiced speech /s/ and its (b) short time spectrum, (c) LP spectrum and (d) inverse spectrum (LP order: 10).

LP residual for voiced and unvoiced segments

LP residual signal is obtained by passing the speech signal through inverse filter designed with LP coefficients (LPCs). The block diagram of the inverse filter is shown in Figure 4.

Figure 4: Inverse filter to obtain LP residual signal from speech signal.

Voiced and unvoiced speech segments and their LP residual signals are shown in Figure 5.

Figure 5: (a) Segment of voiced speech /a/ and its (b) LP residual signal, (c) segment of unvoiced speech /s/ and its (d) LP residual signal (LP order: 10).

Autocorrelation function for voiced/unvoiced speech segments and their LP residuals

• Autocorrelation function of the signal \(x[n]\) is computed as
  $$ R[\tau]=\sum\limits_{n=-\infty}^{+\infty}x[n]x[n+\tau] \qquad(7) $$
• The autocorrelation function for the voiced speech segment and its LP residual signal is shown in FigureÃÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’â€Â ÃƒƒÂ¢Ã¢â€šÂ¬Ã¢â€žÂ¢ÃƒÆ’ƒÃ¢â‚¬Â ÃƒÆ’¢â‚¬â„¢ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’¢â‚¬Â ÃƒƒÃ‚¢Ã¢â€šÂ¬Ã¢â€žÂ¢ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’â€Â ÃƒƒÂ¢Ã¢â€šÂ¬Ã¢â€žÂ¢ÃƒÆ’ƒÃ‚¢Ã¢â€šÂ¬Ã‚ ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’‚¢ÃƒÃ‚¢Ã¢â‚¬Å¡Ã‚¬ÃƒÃ‚¢Ã¢â‚¬Å¾Ã‚¢ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’â€Â ÃƒƒÂ¢Ã¢â€šÂ¬Ã¢â€žÂ¢ÃƒÆ’ƒÃ¢â‚¬Â ÃƒÆ’¢â‚¬â„¢ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’‚¢ÃƒÃ‚¢Ã¢â‚¬Å¡Ã‚¬ÃƒÃ¢â‚¬Å¡Ãƒâ€šÃ‚ ÃƒƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’â€Â ÃƒƒÂ¢Ã¢â€šÂ¬Ã¢â€žÂ¢ÃƒÆ’ƒÃ¢â‚¬Å¡Ãƒâ€šÃ‚¢ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’‚¢ÃƒÃ‚¢Ã¢â€šÂ¬Ã…¡ÃƒÃ¢â‚¬Å¡Ãƒâ€šÃ‚¬ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’‚¢ÃƒÃ‚¢Ã¢â€šÂ¬Ã…¾ÃƒÃ¢â‚¬Å¡Ãƒâ€šÃ‚¢ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’â€Â ÃƒƒÂ¢Ã¢â€šÂ¬Ã¢â€žÂ¢ÃƒÆ’ƒÃ¢â‚¬Â ÃƒÆ’¢â‚¬â„¢ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’¢â‚¬Â ÃƒƒÃ‚¢Ã¢â€šÂ¬Ã¢â€žÂ¢ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’â€Â ÃƒƒÂ¢Ã¢â€šÂ¬Ã¢â€žÂ¢ÃƒÆ’ƒÃ¢â‚¬Å¡Ãƒâ€šÃ‚¢ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’‚¢ÃƒÃ‚¢Ã¢â€šÂ¬Ã…¡ÃƒÃ¢â‚¬Å¡Ãƒâ€šÃ‚¬ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’¢â‚¬Å¡ÃƒÃ¢â‚¬Å¡Ãƒâ€šÃ‚ ÃƒƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’â€Â ÃƒƒÂ¢Ã¢â€šÂ¬Ã¢â€žÂ¢ÃƒÆ’ƒÃ¢â‚¬Â ÃƒÆ’¢â‚¬â„¢ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’¢â‚¬Å¡ÃƒÃ¢â‚¬Å¡Ãƒâ€šÃ‚¢ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’â€Â ÃƒƒÂ¢Ã¢â€šÂ¬Ã¢â€žÂ¢ÃƒÆ’ƒÃ¢â‚¬Å¡Ãƒâ€šÃ‚¢ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’‚¢ÃƒÃ‚¢Ã¢â‚¬Å¡Ã‚¬ÃƒÃ¢â‚¬Â¦Ãƒâ€šÃ‚¡ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’¢â‚¬Å¡ÃƒÃ¢â‚¬Å¡Ãƒâ€šÃ‚¬ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’â€Â ÃƒƒÂ¢Ã¢â€šÂ¬Ã¢â€žÂ¢ÃƒÆ’ƒÃ¢â‚¬Å¡Ãƒâ€šÃ‚¢ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’‚¢ÃƒÃ‚¢Ã¢â‚¬Å¡Ã‚¬ÃƒÃ¢â‚¬Â¦Ãƒâ€šÃ‚¾ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’¢â‚¬Å¡ÃƒÃ¢â‚¬Å¡Ãƒâ€šÃ‚¢ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’â€Â ÃƒƒÂ¢Ã¢â€šÂ¬Ã¢â€žÂ¢ÃƒÆ’ƒÃ¢â‚¬Â ÃƒÆ’¢â‚¬â„¢ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’¢â‚¬Â ÃƒƒÃ‚¢Ã¢â€šÂ¬Ã¢â€žÂ¢ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’â€Â ÃƒƒÂ¢Ã¢â€šÂ¬Ã¢â€žÂ¢ÃƒÆ’ƒÃ‚¢Ã¢â€šÂ¬Ã‚ ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’‚¢ÃƒÃ‚¢Ã¢â‚¬Å¡Ã‚¬ÃƒÃ‚¢Ã¢â‚¬Å¾Ã‚¢ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’â€Â ÃƒƒÂ¢Ã¢â€šÂ¬Ã¢â€žÂ¢ÃƒÆ’ƒÃ¢â‚¬Â ÃƒÆ’¢â‚¬â„¢ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’‚¢ÃƒÃ‚¢Ã¢â‚¬Å¡Ã‚¬ÃƒÃ¢â‚¬Â¦Ãƒâ€šÃ‚¡ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’â€Â ÃƒƒÂ¢Ã¢â€šÂ¬Ã¢â€žÂ¢ÃƒÆ’ƒÃ‚¢Ã¢â€šÂ¬Ã…¡ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’¢â‚¬Å¡ÃƒÃ¢â‚¬Å¡Ãƒâ€šÃ‚¢ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’â€Â ÃƒƒÂ¢Ã¢â€šÂ¬Ã¢â€žÂ¢ÃƒÆ’ƒÃ¢â‚¬Â ÃƒÆ’¢â‚¬â„¢ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’¢â‚¬Â ÃƒƒÃ‚¢Ã¢â€šÂ¬Ã¢â€žÂ¢ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’â€Â ÃƒƒÂ¢Ã¢â€šÂ¬Ã¢â€žÂ¢ÃƒÆ’ƒÃ‚¢Ã¢â€šÂ¬Ã…¡ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’¢â‚¬Å¡ÃƒÃ¢â‚¬Å¡Ãƒâ€šÃ‚¢ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’â€Â ÃƒƒÂ¢Ã¢â€šÂ¬Ã¢â€žÂ¢ÃƒÆ’ƒÃ¢â‚¬Â ÃƒÆ’¢â‚¬â„¢ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’¢â‚¬Å¡ÃƒÃ¢â‚¬Å¡Ãƒâ€šÃ‚¢ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’â€Â ÃƒƒÂ¢Ã¢â€šÂ¬Ã¢â€žÂ¢ÃƒÆ’ƒÃ¢â‚¬Å¡Ãƒâ€šÃ‚¢ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’‚¢ÃƒÃ‚¢Ã¢â‚¬Å¡Ã‚¬ÃƒÃ¢â‚¬Â¦Ãƒâ€šÃ‚¡ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’¢â‚¬Å¡ÃƒÃ¢â‚¬Å¡Ãƒâ€šÃ‚¬ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’â€Â ÃƒƒÂ¢Ã¢â€šÂ¬Ã¢â€žÂ¢ÃƒÆ’ƒÃ‚¢Ã¢â€šÂ¬Ã‚¦ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’¢â‚¬Å¡ÃƒÃ¢â‚¬Å¡Ãƒâ€šÃ‚¡ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’â€Â ÃƒƒÂ¢Ã¢â€šÂ¬Ã¢â€žÂ¢ÃƒÆ’ƒÃ¢â‚¬Â ÃƒÆ’¢â‚¬â„¢ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’‚¢ÃƒÃ‚¢Ã¢â‚¬Å¡Ã‚¬ÃƒÃ¢â‚¬Â¦Ãƒâ€šÃ‚¡ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’â€Â ÃƒƒÂ¢Ã¢â€šÂ¬Ã¢â€žÂ¢ÃƒÆ’ƒÃ‚¢Ã¢â€šÂ¬Ã…¡ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’¢â‚¬Å¡ÃƒÃ¢â‚¬Å¡Ãƒâ€šÃ‚¬ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’â€Â ÃƒƒÂ¢Ã¢â€šÂ¬Ã¢â€žÂ¢ÃƒÆ’ƒÃ¢â‚¬Â ÃƒÆ’¢â‚¬â„¢ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’¢â‚¬Â ÃƒƒÃ‚¢Ã¢â€šÂ¬Ã¢â€žÂ¢ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’â€Â ÃƒƒÂ¢Ã¢â€šÂ¬Ã¢â€žÂ¢ÃƒÆ’ƒÃ¢â‚¬Å¡Ãƒâ€šÃ‚¢ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’‚¢ÃƒÃ‚¢Ã¢â€šÂ¬Ã…¡ÃƒÃ¢â‚¬Å¡Ãƒâ€šÃ‚¬ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’¢â‚¬Å¡ÃƒÃ¢â‚¬Å¡Ãƒâ€šÃ‚¦ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’â€Â ÃƒƒÂ¢Ã¢â€šÂ¬Ã¢â€žÂ¢ÃƒÆ’ƒÃ¢â‚¬Â ÃƒÆ’¢â‚¬â„¢ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’‚¢ÃƒÃ‚¢Ã¢â‚¬Å¡Ã‚¬ÃƒÃ¢â‚¬Â¦Ãƒâ€šÃ‚¡ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’â€Â ÃƒƒÂ¢Ã¢â€šÂ¬Ã¢â€žÂ¢ÃƒÆ’ƒÃ‚¢Ã¢â€šÂ¬Ã…¡ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’¢â‚¬Å¡ÃƒÃ¢â‚¬Å¡Ãƒâ€šÃ‚¡ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’â€Â ÃƒƒÂ¢Ã¢â€šÂ¬Ã¢â€žÂ¢ÃƒÆ’ƒÃ¢â‚¬Â ÃƒÆ’¢â‚¬â„¢ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’¢â‚¬Â ÃƒƒÃ‚¢Ã¢â€šÂ¬Ã¢â€žÂ¢ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’â€Â ÃƒƒÂ¢Ã¢â€šÂ¬Ã¢â€žÂ¢ÃƒÆ’ƒÃ‚¢Ã¢â€šÂ¬Ã‚ ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’‚¢ÃƒÃ‚¢Ã¢â‚¬Å¡Ã‚¬ÃƒÃ‚¢Ã¢â‚¬Å¾Ã‚¢ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’â€Â ÃƒƒÂ¢Ã¢â€šÂ¬Ã¢â€žÂ¢ÃƒÆ’ƒÃ¢â‚¬Â ÃƒÆ’¢â‚¬â„¢ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’¢â‚¬Å¡ÃƒÃ¢â‚¬Å¡Ãƒâ€šÃ‚¢ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’â€Â ÃƒƒÂ¢Ã¢â€šÂ¬Ã¢â€žÂ¢ÃƒÆ’ƒÃ¢â‚¬Å¡Ãƒâ€šÃ‚¢ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’‚¢ÃƒÃ‚¢Ã¢â‚¬Å¡Ã‚¬ÃƒÃ¢â‚¬Â¦Ãƒâ€šÃ‚¡ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’¢â‚¬Å¡ÃƒÃ¢â‚¬Å¡Ãƒâ€šÃ‚¬ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’â€Â ÃƒƒÂ¢Ã¢â€šÂ¬Ã¢â€žÂ¢ÃƒÆ’ƒÃ‚¢Ã¢â€šÂ¬Ã‚¦ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’¢â‚¬Å¡ÃƒÃ¢â‚¬Å¡Ãƒâ€šÃ‚¡ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’â€Â ÃƒƒÂ¢Ã¢â€šÂ¬Ã¢â€žÂ¢ÃƒÆ’ƒÃ¢â‚¬Â ÃƒÆ’¢â‚¬â„¢ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’¢â‚¬Â ÃƒƒÃ‚¢Ã¢â€šÂ¬Ã¢â€žÂ¢ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’â€Â ÃƒƒÂ¢Ã¢â€šÂ¬Ã¢â€žÂ¢ÃƒÆ’ƒÃ¢â‚¬Å¡Ãƒâ€šÃ‚¢ÃÃÆ’ƒÆ’†â€™Ãƒâ€ Ã¢â‚¬â„¢ÃƒÆ’†Ã¢â‚¬â„¢Ãƒƒâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’‚¢ÃƒÃ‚¢Ã¢â€šÂ¬Ã…¡ÃƒÃ¢â‚¬Å¡Ãƒâ€šÃ‚¬ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’¢â‚¬Â¦ÃƒÃ¢â‚¬Å¡Ãƒâ€šÃ‚¡ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’â€Â ÃƒƒÂ¢Ã¢â€šÂ¬Ã¢â€žÂ¢ÃƒÆ’ƒÃ¢â‚¬Â ÃƒÆ’¢â‚¬â„¢ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’‚¢ÃƒÃ‚¢Ã¢â‚¬Å¡Ã‚¬ÃƒÃ¢â‚¬Â¦Ãƒâ€šÃ‚¡ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’â€Â ÃƒƒÂ¢Ã¢â€šÂ¬Ã¢â€žÂ¢ÃƒÆ’ƒÃ‚¢Ã¢â€šÂ¬Ã…¡ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’¢â‚¬Å¡ÃƒÃ¢â‚¬Å¡Ãƒâ€šÃ‚ 6.


Figure 6: (a) Segment of voiced speech /a/ and its (b) autocorrelation function, (c) LP residual for the voiced speech segment and its (d) autocorrelation function (LP order: 10).


• The autocorrelation function for the unvoiced speech segment and its LP residual signal is shown in Figure 7.
  
  

Figure 7: (a) Segment of unvoiced speech /s/ and its (b) autocorrelation function, (c) LP residual for the unvoiced speech segment and its (d) autocorrelation function (LP order: 10).

Glottal pulse shape in voiced portion of a speech signal

• By integrating the LP residual we can obtain the glottal pulse shape, it is also known as glottal volume velocity.
• A segment of voiced speech its LP residual and glottal pulse (glottal volume velocity) waveforms are shown in FigureÃÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’â€Â ÃƒƒÂ¢Ã¢â€šÂ¬Ã¢â€žÂ¢ÃƒÆ’ƒÃ¢â‚¬Â ÃƒÆ’¢â‚¬â„¢ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’¢â‚¬Â ÃƒƒÃ‚¢Ã¢â€šÂ¬Ã¢â€žÂ¢ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’â€Â ÃƒƒÂ¢Ã¢â€šÂ¬Ã¢â€žÂ¢ÃƒÆ’ƒÃ‚¢Ã¢â€šÂ¬Ã‚ ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’‚¢ÃƒÃ‚¢Ã¢â‚¬Å¡Ã‚¬ÃƒÃ‚¢Ã¢â‚¬Å¾Ã‚¢ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’â€Â ÃƒƒÂ¢Ã¢â€šÂ¬Ã¢â€žÂ¢ÃƒÆ’ƒÃ¢â‚¬Â ÃƒÆ’¢â‚¬â„¢ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’‚¢ÃƒÃ‚¢Ã¢â‚¬Å¡Ã‚¬ÃƒÃ¢â‚¬Å¡Ãƒâ€šÃ‚ ÃƒƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’â€Â ÃƒƒÂ¢Ã¢â€šÂ¬Ã¢â€žÂ¢ÃƒÆ’ƒÃ¢â‚¬Å¡Ãƒâ€šÃ‚¢ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’‚¢ÃƒÃ‚¢Ã¢â€šÂ¬Ã…¡ÃƒÃ¢â‚¬Å¡Ãƒâ€šÃ‚¬ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’‚¢ÃƒÃ‚¢Ã¢â€šÂ¬Ã…¾ÃƒÃ¢â‚¬Å¡Ãƒâ€šÃ‚¢ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’â€Â ÃƒƒÂ¢Ã¢â€šÂ¬Ã¢â€žÂ¢ÃƒÆ’ƒÃ¢â‚¬Â ÃƒÆ’¢â‚¬â„¢ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’¢â‚¬Â ÃƒƒÃ‚¢Ã¢â€šÂ¬Ã¢â€žÂ¢ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’â€Â ÃƒƒÂ¢Ã¢â€šÂ¬Ã¢â€žÂ¢ÃƒÆ’ƒÃ¢â‚¬Å¡Ãƒâ€šÃ‚¢ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’‚¢ÃƒÃ‚¢Ã¢â€šÂ¬Ã…¡ÃƒÃ¢â‚¬Å¡Ãƒâ€šÃ‚¬ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’¢â‚¬Å¡ÃƒÃ¢â‚¬Å¡Ãƒâ€šÃ‚ ÃƒƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’â€Â ÃƒƒÂ¢Ã¢â€šÂ¬Ã¢â€žÂ¢ÃƒÆ’ƒÃ¢â‚¬Â ÃƒÆ’¢â‚¬â„¢ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’¢â‚¬Å¡ÃƒÃ¢â‚¬Å¡Ãƒâ€šÃ‚¢ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’â€Â ÃƒƒÂ¢Ã¢â€šÂ¬Ã¢â€žÂ¢ÃƒÆ’ƒÃ¢â‚¬Å¡Ãƒâ€šÃ‚¢ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’‚¢ÃƒÃ‚¢Ã¢â‚¬Å¡Ã‚¬ÃƒÃ¢â‚¬Â¦Ãƒâ€šÃ‚¡ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’¢â‚¬Å¡ÃƒÃ¢â‚¬Å¡Ãƒâ€šÃ‚¬ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’â€Â ÃƒƒÂ¢Ã¢â€šÂ¬Ã¢â€žÂ¢ÃƒÆ’ƒÃ¢â‚¬Å¡Ãƒâ€šÃ‚¢ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’‚¢ÃƒÃ‚¢Ã¢â‚¬Å¡Ã‚¬ÃƒÃ¢â‚¬Â¦Ãƒâ€šÃ‚¾ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’¢â‚¬Å¡ÃƒÃ¢â‚¬Å¡Ãƒâ€šÃ‚¢ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’â€Â ÃƒƒÂ¢Ã¢â€šÂ¬Ã¢â€žÂ¢ÃƒÆ’ƒÃ¢â‚¬Â ÃƒÆ’¢â‚¬â„¢ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’¢â‚¬Â ÃƒƒÃ‚¢Ã¢â€šÂ¬Ã¢â€žÂ¢ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’â€Â ÃƒƒÂ¢Ã¢â€šÂ¬Ã¢â€žÂ¢ÃƒÆ’ƒÃ‚¢Ã¢â€šÂ¬Ã‚ ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’‚¢ÃƒÃ‚¢Ã¢â‚¬Å¡Ã‚¬ÃƒÃ‚¢Ã¢â‚¬Å¾Ã‚¢ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’â€Â ÃƒƒÂ¢Ã¢â€šÂ¬Ã¢â€žÂ¢ÃƒÆ’ƒÃ¢â‚¬Â ÃƒÆ’¢â‚¬â„¢ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’‚¢ÃƒÃ‚¢Ã¢â‚¬Å¡Ã‚¬ÃƒÃ¢â‚¬Â¦Ãƒâ€šÃ‚¡ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’â€Â ÃƒƒÂ¢Ã¢â€šÂ¬Ã¢â€žÂ¢ÃƒÆ’ƒÃ‚¢Ã¢â€šÂ¬Ã…¡ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’¢â‚¬Å¡ÃƒÃ¢â‚¬Å¡Ãƒâ€šÃ‚¢ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’â€Â ÃƒƒÂ¢Ã¢â€šÂ¬Ã¢â€žÂ¢ÃƒÆ’ƒÃ¢â‚¬Â ÃƒÆ’¢â‚¬â„¢ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’¢â‚¬Â ÃƒƒÃ‚¢Ã¢â€šÂ¬Ã¢â€žÂ¢ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’â€Â ÃƒƒÂ¢Ã¢â€šÂ¬Ã¢â€žÂ¢ÃƒÆ’ƒÃ‚¢Ã¢â€šÂ¬Ã…¡ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’¢â‚¬Å¡ÃƒÃ¢â‚¬Å¡Ãƒâ€šÃ‚¢ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’â€Â ÃƒƒÂ¢Ã¢â€šÂ¬Ã¢â€žÂ¢ÃƒÆ’ƒÃ¢â‚¬Â ÃƒÆ’¢â‚¬â„¢ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’¢â‚¬Å¡ÃƒÃ¢â‚¬Å¡Ãƒâ€šÃ‚¢ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’â€Â ÃƒƒÂ¢Ã¢â€šÂ¬Ã¢â€žÂ¢ÃƒÆ’ƒÃ¢â‚¬Å¡Ãƒâ€šÃ‚¢ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’‚¢ÃƒÃ‚¢Ã¢â‚¬Å¡Ã‚¬ÃƒÃ¢â‚¬Â¦Ãƒâ€šÃ‚¡ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’¢â‚¬Å¡ÃƒÃ¢â‚¬Å¡Ãƒâ€šÃ‚¬ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’â€Â ÃƒƒÂ¢Ã¢â€šÂ¬Ã¢â€žÂ¢ÃƒÆ’ƒÃ‚¢Ã¢â€šÂ¬Ã‚¦ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’¢â‚¬Å¡ÃƒÃ¢â‚¬Å¡Ãƒâ€šÃ‚¡ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’â€Â ÃƒƒÂ¢Ã¢â€šÂ¬Ã¢â€žÂ¢ÃƒÆ’ƒÃ¢â‚¬Â ÃƒÆ’¢â‚¬â„¢ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’‚¢ÃƒÃ‚¢Ã¢â‚¬Å¡Ã‚¬ÃƒÃ¢â‚¬Â¦Ãƒâ€šÃ‚¡ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’â€Â ÃƒƒÂ¢Ã¢â€šÂ¬Ã¢â€žÂ¢ÃƒÆ’ƒÃ‚¢Ã¢â€šÂ¬Ã…¡ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’¢â‚¬Å¡ÃƒÃ¢â‚¬Å¡Ãƒâ€šÃ‚¬ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’â€Â ÃƒƒÂ¢Ã¢â€šÂ¬Ã¢â€žÂ¢ÃƒÆ’ƒÃ¢â‚¬Â ÃƒÆ’¢â‚¬â„¢ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’¢â‚¬Â ÃƒƒÃ‚¢Ã¢â€šÂ¬Ã¢â€žÂ¢ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’â€Â ÃƒƒÂ¢Ã¢â€šÂ¬Ã¢â€žÂ¢ÃƒÆ’ƒÃ¢â‚¬Å¡Ãƒâ€šÃ‚¢ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’‚¢ÃƒÃ‚¢Ã¢â€šÂ¬Ã…¡ÃƒÃ¢â‚¬Å¡Ãƒâ€šÃ‚¬ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’¢â‚¬Å¡ÃƒÃ¢â‚¬Å¡Ãƒâ€šÃ‚¦ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’â€Â ÃƒƒÂ¢Ã¢â€šÂ¬Ã¢â€žÂ¢ÃƒÆ’ƒÃ¢â‚¬Â ÃƒÆ’¢â‚¬â„¢ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’‚¢ÃƒÃ‚¢Ã¢â‚¬Å¡Ã‚¬ÃƒÃ¢â‚¬Â¦Ãƒâ€šÃ‚¡ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’â€Â ÃƒƒÂ¢Ã¢â€šÂ¬Ã¢â€žÂ¢ÃƒÆ’ƒÃ‚¢Ã¢â€šÂ¬Ã…¡ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’¢â‚¬Å¡ÃƒÃ¢â‚¬Å¡Ãƒâ€šÃ‚¡ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’â€Â ÃƒƒÂ¢Ã¢â€šÂ¬Ã¢â€žÂ¢ÃƒÆ’ƒÃ¢â‚¬Â ÃƒÆ’¢â‚¬â„¢ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’¢â‚¬Â ÃƒƒÃ‚¢Ã¢â€šÂ¬Ã¢â€žÂ¢ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’â€Â ÃƒƒÂ¢Ã¢â€šÂ¬Ã¢â€žÂ¢ÃƒÆ’ƒÃ‚¢Ã¢â€šÂ¬Ã‚ ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’‚¢ÃƒÃ‚¢Ã¢â‚¬Å¡Ã‚¬ÃƒÃ‚¢Ã¢â‚¬Å¾Ã‚¢ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’â€Â ÃƒƒÂ¢Ã¢â€šÂ¬Ã¢â€žÂ¢ÃƒÆ’ƒÃ¢â‚¬Â ÃƒÆ’¢â‚¬â„¢ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’¢â‚¬Å¡ÃƒÃ¢â‚¬Å¡Ãƒâ€šÃ‚¢ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’â€Â ÃƒƒÂ¢Ã¢â€šÂ¬Ã¢â€žÂ¢ÃƒÆ’ƒÃ¢â‚¬Å¡Ãƒâ€šÃ‚¢ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’‚¢ÃƒÃ‚¢Ã¢â‚¬Å¡Ã‚¬ÃƒÃ¢â‚¬Â¦Ãƒâ€šÃ‚¡ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’¢â‚¬Å¡ÃƒÃ¢â‚¬Å¡Ãƒâ€šÃ‚¬ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’â€Â ÃƒƒÂ¢Ã¢â€šÂ¬Ã¢â€žÂ¢ÃƒÆ’ƒÃ‚¢Ã¢â€šÂ¬Ã‚¦ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’¢â‚¬Å¡ÃƒÃ¢â‚¬Å¡Ãƒâ€šÃ‚¡ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’â€Â ÃƒƒÂ¢Ã¢â€šÂ¬Ã¢â€žÂ¢ÃƒÆ’ƒÃ¢â‚¬Â ÃƒÆ’¢â‚¬â„¢ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’¢â‚¬Â ÃƒƒÃ‚¢Ã¢â€šÂ¬Ã¢â€žÂ¢ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’â€Â ÃƒƒÂ¢Ã¢â€šÂ¬Ã¢â€žÂ¢ÃƒÆ’ƒÃ¢â‚¬Å¡Ãƒâ€šÃ‚¢ÃÆâ€ââ€žÂ¢ÃƒÆ’ƒâ€ Ã¢â‚¬â„¢ÃƒÆ’†â€™Ãƒâ€ ÃƒÂ¢Ã¢â€šÂ¬Ã¢â€žÂ¢ÃƒÆ’ƒâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’‚¢ÃƒÃ‚¢Ã¢â€šÂ¬Ã…¡ÃƒÃ¢â‚¬Å¡Ãƒâ€šÃ‚¬ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’¢â‚¬Â¦ÃƒÃ¢â‚¬Å¡Ãƒâ€šÃ‚¡ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’â€Â ÃƒƒÂ¢Ã¢â€šÂ¬Ã¢â€žÂ¢ÃƒÆ’ƒÃ¢â‚¬Â ÃƒÆ’¢â‚¬â„¢ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’‚¢ÃƒÃ‚¢Ã¢â‚¬Å¡Ã‚¬ÃƒÃ¢â‚¬Â¦Ãƒâ€šÃ‚¡ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’â€Â ÃƒƒÂ¢Ã¢â€šÂ¬Ã¢â€žÂ¢ÃƒÆ’ƒÃ‚¢Ã¢â€šÂ¬Ã…¡ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’¢â‚¬Å¡ÃƒÃ¢â‚¬Å¡Ãƒâ€šÃ‚ 8.

Figure 8: (a) Segment of voiced speech /a/, its (b) LP residual and (c) glottal pulse waveform (LP order: 10).

LP spectrum for different LP orders

• Compute LPCs for different LP orders (14, 10, 6, 3 and 1), and compute LP spectrum for each set of LPCs.
• A segment of voiced speech and its LP spectrum for different LP orders (14, 10, 6, 3 and 1) are shown in FigureÃÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’â€Â ÃƒƒÂ¢Ã¢â€šÂ¬Ã¢â€žÂ¢ÃƒÆ’ƒÃ¢â‚¬Â ÃƒÆ’¢â‚¬â„¢ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’¢â‚¬Â ÃƒƒÃ‚¢Ã¢â€šÂ¬Ã¢â€žÂ¢ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’â€Â ÃƒƒÂ¢Ã¢â€šÂ¬Ã¢â€žÂ¢ÃƒÆ’ƒÃ‚¢Ã¢â€šÂ¬Ã‚ ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’‚¢ÃƒÃ‚¢Ã¢â‚¬Å¡Ã‚¬ÃƒÃ‚¢Ã¢â‚¬Å¾Ã‚¢ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’â€Â ÃƒƒÂ¢Ã¢â€šÂ¬Ã¢â€žÂ¢ÃƒÆ’ƒÃ¢â‚¬Â ÃƒÆ’¢â‚¬â„¢ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’‚¢ÃƒÃ‚¢Ã¢â‚¬Å¡Ã‚¬ÃƒÃ¢â‚¬Å¡Ãƒâ€šÃ‚ ÃƒƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’â€Â ÃƒƒÂ¢Ã¢â€šÂ¬Ã¢â€žÂ¢ÃƒÆ’ƒÃ¢â‚¬Å¡Ãƒâ€šÃ‚¢ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’‚¢ÃƒÃ‚¢Ã¢â€šÂ¬Ã…¡ÃƒÃ¢â‚¬Å¡Ãƒâ€šÃ‚¬ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’‚¢ÃƒÃ‚¢Ã¢â€šÂ¬Ã…¾ÃƒÃ¢â‚¬Å¡Ãƒâ€šÃ‚¢ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’â€Â ÃƒƒÂ¢Ã¢â€šÂ¬Ã¢â€žÂ¢ÃƒÆ’ƒÃ¢â‚¬Â ÃƒÆ’¢â‚¬â„¢ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’¢â‚¬Â ÃƒƒÃ‚¢Ã¢â€šÂ¬Ã¢â€žÂ¢ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’â€Â ÃƒƒÂ¢Ã¢â€šÂ¬Ã¢â€žÂ¢ÃƒÆ’ƒÃ¢â‚¬Å¡Ãƒâ€šÃ‚¢ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’‚¢ÃƒÃ‚¢Ã¢â€šÂ¬Ã…¡ÃƒÃ¢â‚¬Å¡Ãƒâ€šÃ‚¬ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’¢â‚¬Å¡ÃƒÃ¢â‚¬Å¡Ãƒâ€šÃ‚ ÃƒƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’â€Â ÃƒƒÂ¢Ã¢â€šÂ¬Ã¢â€žÂ¢ÃƒÆ’ƒÃ¢â‚¬Â ÃƒÆ’¢â‚¬â„¢ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’¢â‚¬Å¡ÃƒÃ¢â‚¬Å¡Ãƒâ€šÃ‚¢ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’â€Â ÃƒƒÂ¢Ã¢â€šÂ¬Ã¢â€žÂ¢ÃƒÆ’ƒÃ¢â‚¬Å¡Ãƒâ€šÃ‚¢ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’‚¢ÃƒÃ‚¢Ã¢â‚¬Å¡Ã‚¬ÃƒÃ¢â‚¬Â¦Ãƒâ€šÃ‚¡ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’¢â‚¬Å¡ÃƒÃ¢â‚¬Å¡Ãƒâ€šÃ‚¬ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’â€Â ÃƒƒÂ¢Ã¢â€šÂ¬Ã¢â€žÂ¢ÃƒÆ’ƒÃ¢â‚¬Å¡Ãƒâ€šÃ‚¢ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’‚¢ÃƒÃ‚¢Ã¢â‚¬Å¡Ã‚¬ÃƒÃ¢â‚¬Â¦Ãƒâ€šÃ‚¾ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’¢â‚¬Å¡ÃƒÃ¢â‚¬Å¡Ãƒâ€šÃ‚¢ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’â€Â ÃƒƒÂ¢Ã¢â€šÂ¬Ã¢â€žÂ¢ÃƒÆ’ƒÃ¢â‚¬Â ÃƒÆ’¢â‚¬â„¢ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’¢â‚¬Â ÃƒƒÃ‚¢Ã¢â€šÂ¬Ã¢â€žÂ¢ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’â€Â ÃƒƒÂ¢Ã¢â€šÂ¬Ã¢â€žÂ¢ÃƒÆ’ƒÃ‚¢Ã¢â€šÂ¬Ã‚ ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’‚¢ÃƒÃ‚¢Ã¢â‚¬Å¡Ã‚¬ÃƒÃ‚¢Ã¢â‚¬Å¾Ã‚¢ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’â€Â ÃƒƒÂ¢Ã¢â€šÂ¬Ã¢â€žÂ¢ÃƒÆ’ƒÃ¢â‚¬Â ÃƒÆ’¢â‚¬â„¢ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’‚¢ÃƒÃ‚¢Ã¢â‚¬Å¡Ã‚¬ÃƒÃ¢â‚¬Â¦Ãƒâ€šÃ‚¡ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’â€Â ÃƒƒÂ¢Ã¢â€šÂ¬Ã¢â€žÂ¢ÃƒÆ’ƒÃ‚¢Ã¢â€šÂ¬Ã…¡ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’¢â‚¬Å¡ÃƒÃ¢â‚¬Å¡Ãƒâ€šÃ‚¢ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’â€Â ÃƒƒÂ¢Ã¢â€šÂ¬Ã¢â€žÂ¢ÃƒÆ’ƒÃ¢â‚¬Â ÃƒÆ’¢â‚¬â„¢ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’¢â‚¬Â ÃƒƒÃ‚¢Ã¢â€šÂ¬Ã¢â€žÂ¢ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’â€Â ÃƒƒÂ¢Ã¢â€šÂ¬Ã¢â€žÂ¢ÃƒÆ’ƒÃ‚¢Ã¢â€šÂ¬Ã…¡ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’¢â‚¬Å¡ÃƒÃ¢â‚¬Å¡Ãƒâ€šÃ‚¢ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’â€Â ÃƒƒÂ¢Ã¢â€šÂ¬Ã¢â€žÂ¢ÃƒÆ’ƒÃ¢â‚¬Â ÃƒÆ’¢â‚¬â„¢ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’¢â‚¬Å¡ÃƒÃ¢â‚¬Å¡Ãƒâ€šÃ‚¢ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’â€Â ÃƒƒÂ¢Ã¢â€šÂ¬Ã¢â€žÂ¢ÃƒÆ’ƒÃ¢â‚¬Å¡Ãƒâ€šÃ‚¢ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’‚¢ÃƒÃ‚¢Ã¢â‚¬Å¡Ã‚¬ÃƒÃ¢â‚¬Â¦Ãƒâ€šÃ‚¡ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’¢â‚¬Å¡ÃƒÃ¢â‚¬Å¡Ãƒâ€šÃ‚¬ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’â€Â ÃƒƒÂ¢Ã¢â€šÂ¬Ã¢â€žÂ¢ÃƒÆ’ƒÃ‚¢Ã¢â€šÂ¬Ã‚¦ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’¢â‚¬Å¡ÃƒÃ¢â‚¬Å¡Ãƒâ€šÃ‚¡ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’â€Â ÃƒƒÂ¢Ã¢â€šÂ¬Ã¢â€žÂ¢ÃƒÆ’ƒÃ¢â‚¬Â ÃƒÆ’¢â‚¬â„¢ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’‚¢ÃƒÃ‚¢Ã¢â‚¬Å¡Ã‚¬ÃƒÃ¢â‚¬Â¦Ãƒâ€šÃ‚¡ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’â€Â ÃƒƒÂ¢Ã¢â€šÂ¬Ã¢â€žÂ¢ÃƒÆ’ƒÃ‚¢Ã¢â€šÂ¬Ã…¡ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’¢â‚¬Å¡ÃƒÃ¢â‚¬Å¡Ãƒâ€šÃ‚¬ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’â€Â ÃƒƒÂ¢Ã¢â€šÂ¬Ã¢â€žÂ¢ÃƒÆ’ƒÃ¢â‚¬Â ÃƒÆ’¢â‚¬â„¢ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’¢â‚¬Â ÃƒƒÃ‚¢Ã¢â€šÂ¬Ã¢â€žÂ¢ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’â€Â ÃƒƒÂ¢Ã¢â€šÂ¬Ã¢â€žÂ¢ÃƒÆ’ƒÃ¢â‚¬Å¡Ãƒâ€šÃ‚¢ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’‚¢ÃƒÃ‚¢Ã¢â€šÂ¬Ã…¡ÃƒÃ¢â‚¬Å¡Ãƒâ€šÃ‚¬ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’¢â‚¬Å¡ÃƒÃ¢â‚¬Å¡Ãƒâ€šÃ‚¦ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’â€Â ÃƒƒÂ¢Ã¢â€šÂ¬Ã¢â€žÂ¢ÃƒÆ’ƒÃ¢â‚¬Â ÃƒÆ’¢â‚¬â„¢ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’‚¢ÃƒÃ‚¢Ã¢â‚¬Å¡Ã‚¬ÃƒÃ¢â‚¬Â¦Ãƒâ€šÃ‚¡ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’â€Â ÃƒƒÂ¢Ã¢â€šÂ¬Ã¢â€žÂ¢ÃƒÆ’ƒÃ‚¢Ã¢â€šÂ¬Ã…¡ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’¢â‚¬Å¡ÃƒÃ¢â‚¬Å¡Ãƒâ€šÃ‚¡ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’â€Â ÃƒƒÂ¢Ã¢â€šÂ¬Ã¢â€žÂ¢ÃƒÆ’ƒÃ¢â‚¬Â ÃƒÆ’¢â‚¬â„¢ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’¢â‚¬Â ÃƒƒÃ‚¢Ã¢â€šÂ¬Ã¢â€žÂ¢ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’â€Â ÃƒƒÂ¢Ã¢â€šÂ¬Ã¢â€žÂ¢ÃƒÆ’ƒÃ‚¢Ã¢â€šÂ¬Ã‚ ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’‚¢ÃƒÃ‚¢Ã¢â‚¬Å¡Ã‚¬ÃƒÃ‚¢Ã¢â‚¬Å¾Ã‚¢ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’â€Â ÃƒƒÂ¢Ã¢â€šÂ¬Ã¢â€žÂ¢ÃƒÆ’ƒÃ¢â‚¬Â ÃƒÆ’¢â‚¬â„¢ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’¢â‚¬Å¡ÃƒÃ¢â‚¬Å¡Ãƒâ€šÃ‚¢ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’â€Â ÃƒƒÂ¢Ã¢â€šÂ¬Ã¢â€žÂ¢ÃƒÆ’ƒÃ¢â‚¬Å¡Ãƒâ€šÃ‚¢ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’‚¢ÃƒÃ‚¢Ã¢â‚¬Å¡Ã‚¬ÃƒÃ¢â‚¬Â¦Ãƒâ€šÃ‚¡ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’¢â‚¬Å¡ÃƒÃ¢â‚¬Å¡Ãƒâ€šÃ‚¬ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’â€Â ÃƒƒÂ¢Ã¢â€šÂ¬Ã¢â€žÂ¢ÃƒÆ’ƒÃ‚¢Ã¢â€šÂ¬Ã‚¦ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’¢â‚¬Å¡ÃƒÃ¢â‚¬Å¡Ãƒâ€šÃ‚¡ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’â€Â ÃƒƒÂ¢Ã¢â€šÂ¬Ã¢â€žÂ¢ÃƒÆ’ƒÃ¢â‚¬Â ÃƒÆ’¢â‚¬â„¢ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’¢â‚¬Â ÃƒƒÃ‚¢Ã¢â€šÂ¬Ã¢â€žÂ¢ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’â€Â ÃƒƒÂ¢Ã¢â€šÂ¬Ã¢â€žÂ¢ÃƒÆ’ƒÃ¢â‚¬Å¡Ãƒâ€šÃ‚¢ÃÆâ€â„¢ÃƒÆ’†â€™Ãƒâ€ Ã¢â‚¬â„¢ÃƒÆ’†Ã¢â‚¬â„¢Ãƒƒâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’‚¢ÃƒÃ‚¢Ã¢â€šÂ¬Ã…¡ÃƒÃ¢â‚¬Å¡Ãƒâ€šÃ‚¬ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’¢â‚¬Â¦ÃƒÃ¢â‚¬Å¡Ãƒâ€šÃ‚¡ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’â€Â ÃƒƒÂ¢Ã¢â€šÂ¬Ã¢â€žÂ¢ÃƒÆ’ƒÃ¢â‚¬Â ÃƒÆ’¢â‚¬â„¢ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’‚¢ÃƒÃ‚¢Ã¢â‚¬Å¡Ã‚¬ÃƒÃ¢â‚¬Â¦Ãƒâ€šÃ‚¡ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’â€Â ÃƒƒÂ¢Ã¢â€šÂ¬Ã¢â€žÂ¢ÃƒÆ’ƒÃ‚¢Ã¢â€šÂ¬Ã…¡ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’¢â‚¬Å¡ÃƒÃ¢â‚¬Å¡Ãƒâ€šÃ‚ 9.

Figure 9: (a) Segment of voiced speech /a/, its LP spectrum for the LP order (b) 14, (c) 10, (d) 6, (e) 3 and (f) 1

Normalized error for different LP orders for voiced/unvoiced speech segments

• Normalized error is obtained by normalizing the LP residual energy with respect to speech signal energy. $$\eta = \frac{E_r}{E_s}, \qquad(8) $$ where \(E_r=\sum_{n=0}^{N-1}e^2[n]\) and \(E_s=\sum_{n=0}^{N-1}s^2[n]\) denote the residual and signal energies, respectively.
• Normalized error plots for voiced and unvoiced segments of speech for different LP orders are shown in Figure 10

Figure 10: Normalized error for voiced and unvoiced speech segments for different LP orders.

• Select one of the provided speech utterances and click on the load button.

• Select a short segment (20-50 ms) of voiced speech (say a vowel), using 'Zoom to selection' button, and compute the short-time and LP spectrum by clicking on the spectrum button.

• The windowed signal, the log spectrum and the LP log spectrum are displayed.

• Increase the LP order from the default value 1, to values provided in the drop down box. Click on the spectrum button to recompute the spectra.

• Observe that the LP log spectrum approximates the short-time spectral envelope (i.e., the formants) better with increases LP order.

• As the LP order is increased to a very large value, say 40, the LP spectrum tries to approximate the short-time spectral envelope more closely thereby approximating peaks due to pitch harmonics.

• Study the effect of preemphasis flag on the LP spectrum.

• Compute the LP residual signal for different LP orders by clicking on the residual button.

• The correlations in the input segment of speech can be seen to be gradually removed in the LP residual signal as the LP order is increased.

• The LP residual signal for an LP order of 10 or above removes most of the correlations in the input signal, and the resulting signal looks like a train of impulses.

• Repeat the experiment by selecting an unvoiced segment of speech, say a fricative /s/.

• Write a brief note on the observations.

Note: Speech segments are sampled at 8KHz (\(F_s = 8000 Hz\)).

• Short time spectrum gives both source and system information. The envelope of the spectrum gives system information (i.e., resonances in terms of formant frequencies) and spectral ripples (fine variations) give source information (i.e., pitch harmonics). It is a real and even function of 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• The linear prediction (LP) analysis models the vocal tract system. LP spectrum is observed to be an envelope of short time spectrum (smoothed version of short time spectrum) and the peaks in LP spectrum indicate the formant frequencies (resonances) of the vocal tract system. With observation it is evident that the LP spectrum is derived from an all-pole filter.

• The inverse spectrum is observed to be reciprocal of the LP spectrum. Therefore, we can observe the valleys corresponding to the peaks in LP spectrum. It is represented by an all-zero filter.

• For voiced speech segment its LP residual is observed to be periodic. In LP residual signal, peak amplitudes refers to closure of vocal folds (glottal closure), where the prediction is poor therefore its results as maximum error.

  
  

• The periodicity in LP residual also indicate the pitch information.

• LP residual is a result of passing the speech signal through inverse filter (i.e., removing the vocal tract information). This is also considered to be the excitation signal (source information).

• LP residual for unvoiced speech segment looks like random noise. This is because the unvoiced speech signal has no periodicity and looks like random noise (no relations among the samples).

• The basic property of the autocorrelation function (even symmetry) is evident in all (voiced/unvoiced speech and LP residual signals) the plots.

  
  

• The samples in a voiced speech segment are highly correlated, therefore we will observe the peaks other than center are also prominent. As voiced speech is periodic, it is inherited in its autocorrelation function also.

  
  

• In LP residual, the correlation among the samples is less, therefore its autocorrelation function contains the peaks at pitch rate. Hence autocorrelation function of a LP residual is useful for pitch computation.

  
  

• The autocorrelation function of an unvoiced speech segment shows a major peak at the center and other peaks are not significant, since unvoiced speech signal appears like random signal.

• The autocorrelation function for the LP residual of an unvoiced speech segment shows a dominant peak at the center and no other peaks in rest of the portion. This is because, unvoiced speech itself looks like random (no correlation among the samples) and its residual reflects still random.

  
  

• Glottal pulse shape shows the change in volume of air. It is also referred to as glottal volume velocity.

  
  

• From the glottal pulse waveform, it is observed that volume of air and its pressure will be maximum at the instant of closure. Following this is the opening of the vocal folds, as a result of which the air pressure decreases.

• LP order determines to some extent the accuracy with which speech production mechanism is modeled.

  
  

• LP analysis uses an all-pole model to characterize the vocal tract system by capturing the resonances with spectrum and source information with LP residual (inverse filter i.e., all-zero filter).

  
  

• LP order determines the number of resonances that can be captured by the model. The maximum number of resonances captured by the model with LP order P is P/2.

  
  

• The length of the vocal tract from glottis to lips is approximately 17 cm. This can generate four to five prominent resonances in 0-4 KHz range. These resonances can be captured with the LP model of order 10. We also should take care of radiation and windowing effects. Therefore with LP order 10-14 we can model the system by capturing required resonances.ÃÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’â€Â ÃƒƒÂ¢Ã¢â€šÂ¬Ã¢â€žÂ¢ÃƒÆ’ƒÃ¢â‚¬Â ÃƒÆ’¢â‚¬â„¢ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’¢â‚¬Â ÃƒƒÃ‚¢Ã¢â€šÂ¬Ã¢â€žÂ¢ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’â€Â ÃƒƒÂ¢Ã¢â€šÂ¬Ã¢â€žÂ¢ÃƒÆ’ƒÃ‚¢Ã¢â€šÂ¬Ã‚ ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’‚¢ÃƒÃ‚¢Ã¢â‚¬Å¡Ã‚¬ÃƒÃ‚¢Ã¢â‚¬Å¾Ã‚¢ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’â€Â ÃƒƒÂ¢Ã¢â€šÂ¬Ã¢â€žÂ¢ÃƒÆ’ƒÃ¢â‚¬Â ÃƒÆ’¢â‚¬â„¢ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’‚¢ÃƒÃ‚¢Ã¢â‚¬Å¡Ã‚¬ÃƒÃ¢â‚¬Å¡Ãƒâ€šÃ‚ ÃƒƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’â€Â ÃƒƒÂ¢Ã¢â€šÂ¬Ã¢â€žÂ¢ÃƒÆ’ƒÃ¢â‚¬Å¡Ãƒâ€šÃ‚¢ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’‚¢ÃƒÃ‚¢Ã¢â€šÂ¬Ã…¡ÃƒÃ¢â‚¬Å¡Ãƒâ€šÃ‚¬ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’‚¢ÃƒÃ‚¢Ã¢â€šÂ¬Ã…¾ÃƒÃ¢â‚¬Å¡Ãƒâ€šÃ‚¢ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’â€Â ÃƒƒÂ¢Ã¢â€šÂ¬Ã¢â€žÂ¢ÃƒÆ’ƒÃ¢â‚¬Â ÃƒÆ’¢â‚¬â„¢ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’¢â‚¬Â ÃƒƒÃ‚¢Ã¢â€šÂ¬Ã¢â€žÂ¢ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’â€Â ÃƒƒÂ¢Ã¢â€šÂ¬Ã¢â€žÂ¢ÃƒÆ’ƒÃ¢â‚¬Å¡Ãƒâ€šÃ‚¢ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’‚¢ÃƒÃ‚¢Ã¢â€šÂ¬Ã…¡ÃƒÃ¢â‚¬Å¡Ãƒâ€šÃ‚¬ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’¢â‚¬Å¡ÃƒÃ¢â‚¬Å¡Ãƒâ€šÃ‚ ÃƒƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’â€Â ÃƒƒÂ¢Ã¢â€šÂ¬Ã¢â€žÂ¢ÃƒÆ’ƒÃ¢â‚¬Â ÃƒÆ’¢â‚¬â„¢ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’¢â‚¬Å¡ÃƒÃ¢â‚¬Å¡Ãƒâ€šÃ‚¢ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’â€Â ÃƒƒÂ¢Ã¢â€šÂ¬Ã¢â€žÂ¢ÃƒÆ’ƒÃ¢â‚¬Å¡Ãƒâ€šÃ‚¢ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’‚¢ÃƒÃ‚¢Ã¢â‚¬Å¡Ã‚¬ÃƒÃ¢â‚¬Â¦Ãƒâ€šÃ‚¡ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’¢â‚¬Å¡ÃƒÃ¢â‚¬Å¡Ãƒâ€šÃ‚¬ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’â€Â ÃƒƒÂ¢Ã¢â€šÂ¬Ã¢â€žÂ¢ÃƒÆ’ƒÃ¢â‚¬Å¡Ãƒâ€šÃ‚¢ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’‚¢ÃƒÃ‚¢Ã¢â‚¬Å¡Ã‚¬ÃƒÃ¢â‚¬Â¦Ãƒâ€šÃ‚¾ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’¢â‚¬Å¡ÃƒÃ¢â‚¬Å¡Ãƒâ€šÃ‚¢ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’â€Â ÃƒƒÂ¢Ã¢â€šÂ¬Ã¢â€žÂ¢ÃƒÆ’ƒÃ¢â‚¬Â ÃƒÆ’¢â‚¬â„¢ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’¢â‚¬Â ÃƒƒÃ‚¢Ã¢â€šÂ¬Ã¢â€žÂ¢ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’â€Â ÃƒƒÂ¢Ã¢â€šÂ¬Ã¢â€žÂ¢ÃƒÆ’ƒÃ‚¢Ã¢â€šÂ¬Ã‚ ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’‚¢ÃƒÃ‚¢Ã¢â‚¬Å¡Ã‚¬ÃƒÃ‚¢Ã¢â‚¬Å¾Ã‚¢ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’â€Â ÃƒƒÂ¢Ã¢â€šÂ¬Ã¢â€žÂ¢ÃƒÆ’ƒÃ¢â‚¬Â ÃƒÆ’¢â‚¬â„¢ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’‚¢ÃƒÃ‚¢Ã¢â‚¬Å¡Ã‚¬ÃƒÃ¢â‚¬Â¦Ãƒâ€šÃ‚¡ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’â€Â ÃƒƒÂ¢Ã¢â€šÂ¬Ã¢â€žÂ¢ÃƒÆ’ƒÃ‚¢Ã¢â€šÂ¬Ã…¡ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’¢â‚¬Å¡ÃƒÃ¢â‚¬Å¡Ãƒâ€šÃ‚¢ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’â€Â ÃƒƒÂ¢Ã¢â€šÂ¬Ã¢â€žÂ¢ÃƒÆ’ƒÃ¢â‚¬Â ÃƒÆ’¢â‚¬â„¢ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’¢â‚¬Â ÃƒƒÃ‚¢Ã¢â€šÂ¬Ã¢â€žÂ¢ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’â€Â ÃƒƒÂ¢Ã¢â€šÂ¬Ã¢â€žÂ¢ÃƒÆ’ƒÃ‚¢Ã¢â€šÂ¬Ã…¡ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’¢â‚¬Å¡ÃƒÃ¢â‚¬Å¡Ãƒâ€šÃ‚¢ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’â€Â ÃƒƒÂ¢Ã¢â€šÂ¬Ã¢â€žÂ¢ÃƒÆ’ƒÃ¢â‚¬Â ÃƒÆ’¢â‚¬â„¢ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’¢â‚¬Å¡ÃƒÃ¢â‚¬Å¡Ãƒâ€šÃ‚¢ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’â€Â ÃƒƒÂ¢Ã¢â€šÂ¬Ã¢â€žÂ¢ÃƒÆ’ƒÃ¢â‚¬Å¡Ãƒâ€šÃ‚¢ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’‚¢ÃƒÃ‚¢Ã¢â‚¬Å¡Ã‚¬ÃƒÃ¢â‚¬Â¦Ãƒâ€šÃ‚¡ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’¢â‚¬Å¡ÃƒÃ¢â‚¬Å¡Ãƒâ€šÃ‚¬ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’â€Â ÃƒƒÂ¢Ã¢â€šÂ¬Ã¢â€žÂ¢ÃƒÆ’ƒÃ‚¢Ã¢â€šÂ¬Ã‚¦ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’¢â‚¬Å¡ÃƒÃ¢â‚¬Å¡Ãƒâ€šÃ‚¡ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’â€Â ÃƒƒÂ¢Ã¢â€šÂ¬Ã¢â€žÂ¢ÃƒÆ’ƒÃ¢â‚¬Â ÃƒÆ’¢â‚¬â„¢ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’‚¢ÃƒÃ‚¢Ã¢â‚¬Å¡Ã‚¬ÃƒÃ¢â‚¬Â¦Ãƒâ€šÃ‚¡ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’â€Â ÃƒƒÂ¢Ã¢â€šÂ¬Ã¢â€žÂ¢ÃƒÆ’ƒÃ‚¢Ã¢â€šÂ¬Ã…¡ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’¢â‚¬Å¡ÃƒÃ¢â‚¬Å¡Ãƒâ€šÃ‚¬ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’â€Â ÃƒƒÂ¢Ã¢â€šÂ¬Ã¢â€žÂ¢ÃƒÆ’ƒÃ¢â‚¬Â ÃƒÆ’¢â‚¬â„¢ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’¢â‚¬Â ÃƒƒÃ‚¢Ã¢â€šÂ¬Ã¢â€žÂ¢ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’â€Â ÃƒƒÂ¢Ã¢â€šÂ¬Ã¢â€žÂ¢ÃƒÆ’ƒÃ¢â‚¬Å¡Ãƒâ€šÃ‚¢ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’‚¢ÃƒÃ‚¢Ã¢â€šÂ¬Ã…¡ÃƒÃ¢â‚¬Å¡Ãƒâ€šÃ‚¬ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’¢â‚¬Å¡ÃƒÃ¢â‚¬Å¡Ãƒâ€šÃ‚¦ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’â€Â ÃƒƒÂ¢Ã¢â€šÂ¬Ã¢â€žÂ¢ÃƒÆ’ƒÃ¢â‚¬Â ÃƒÆ’¢â‚¬â„¢ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’‚¢ÃƒÃ‚¢Ã¢â‚¬Å¡Ã‚¬ÃƒÃ¢â‚¬Â¦Ãƒâ€šÃ‚¡ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’â€Â ÃƒƒÂ¢Ã¢â€šÂ¬Ã¢â€žÂ¢ÃƒÆ’ƒÃ‚¢Ã¢â€šÂ¬Ã…¡ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’¢â‚¬Å¡ÃƒÃ¢â‚¬Å¡Ãƒâ€šÃ‚¡ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’â€Â ÃƒƒÂ¢Ã¢â€šÂ¬Ã¢â€žÂ¢ÃƒÆ’ƒÃ¢â‚¬Â ÃƒÆ’¢â‚¬â„¢ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’¢â‚¬Â ÃƒƒÃ‚¢Ã¢â€šÂ¬Ã¢â€žÂ¢ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’â€Â ÃƒƒÂ¢Ã¢â€šÂ¬Ã¢â€žÂ¢ÃƒÆ’ƒÃ‚¢Ã¢â€šÂ¬Ã‚ ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’‚¢ÃƒÃ‚¢Ã¢â‚¬Å¡Ã‚¬ÃƒÃ‚¢Ã¢â‚¬Å¾Ã‚¢ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’â€Â ÃƒƒÂ¢Ã¢â€šÂ¬Ã¢â€žÂ¢ÃƒÆ’ƒÃ¢â‚¬Â ÃƒÆ’¢â‚¬â„¢ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’¢â‚¬Å¡ÃƒÃ¢â‚¬Å¡Ãƒâ€šÃ‚¢ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’â€Â ÃƒƒÂ¢Ã¢â€šÂ¬Ã¢â€žÂ¢ÃƒÆ’ƒÃ¢â‚¬Å¡Ãƒâ€šÃ‚¢ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’‚¢ÃƒÃ‚¢Ã¢â‚¬Å¡Ã‚¬ÃƒÃ¢â‚¬Â¦Ãƒâ€šÃ‚¡ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’¢â‚¬Å¡ÃƒÃ¢â‚¬Å¡Ãƒâ€šÃ‚¬ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’â€Â ÃƒƒÂ¢Ã¢â€šÂ¬Ã¢â€žÂ¢ÃƒÆ’ƒÃ‚¢Ã¢â€šÂ¬Ã‚¦ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’¢â‚¬Å¡ÃƒÃ¢â‚¬Å¡Ãƒâ€šÃ‚¡ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’â€Â ÃƒƒÂ¢Ã¢â€šÂ¬Ã¢â€žÂ¢ÃƒÆ’ƒÃ¢â‚¬Â ÃƒÆ’¢â‚¬â„¢ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’¢â‚¬Â ÃƒƒÃ‚¢Ã¢â€šÂ¬Ã¢â€žÂ¢ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’â€Â ÃƒƒÂ¢Ã¢â€šÂ¬Ã¢â€žÂ¢ÃƒÆ’ƒÃ¢â‚¬Å¡Ãƒâ€šÃ‚ÂÃâââÃ¢â€šÂ¬Ã…¡Ã‚¬Ã…¡Ãƒâ€šÃ‚¢ÃƒÆ’ƒÆ’ƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’‚¢ÃƒÃ‚¢Ã¢â€šÂ¬Ã…¡ÃƒÃ¢â‚¬Å¡Ãƒâ€šÃ‚¬ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’¢â‚¬Â¦ÃƒÃ¢â‚¬Å¡Ãƒâ€šÃ‚¡ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’â€Â ÃƒƒÂ¢Ã¢â€šÂ¬Ã¢â€žÂ¢ÃƒÆ’ƒÃ¢â‚¬Â ÃƒÆ’¢â‚¬â„¢ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’‚¢ÃƒÃ‚¢Ã¢â‚¬Å¡Ã‚¬ÃƒÃ¢â‚¬Â¦Ãƒâ€šÃ‚¡ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’â€Â ÃƒƒÂ¢Ã¢â€šÂ¬Ã¢â€žÂ¢ÃƒÆ’ƒÃ‚¢Ã¢â€šÂ¬Ã…¡ÃƒÃâ€Â ÃƒÆ’¢â‚¬â„¢ÃƒÆ’¢â‚¬Å¡ÃƒÃ¢â‚¬Å¡Ãƒâ€šÃ‚ 

• System with LP order more than 14 will introduce the spurious resonances, which leads improper representation of the vocal tract system.

1. What is the minimum number of speech samples required to compute the LP coefficients of order p?
2. Explain the differences between autocorrelation and autcovariance formulations of LP analysis? Which is better and why?
3. Suggest an algorithm for voiced/non-voiced region separation using:
   1. LP residual energy
   2. Periodicity information in the LP residual
4. Write an Octave/Scilab program that implements the above algorithms of 3 (1) and 3 (2)

• Digital Processing of Speech Signals, L.R. Rabiner and L.R. Schafer, Chapter 8

• Discrete-Time Speech Signal Processing, Thomas F. Quatieri , Chapter 5